ELEG 3124 SYSTEMS AND SIGNALS Lecture Notes

Department of Electrical EngineeringUniversity of Arkansas

ELEG 3124 SYSTEMS AND SIGNALS

Lecture Notes

Dr. Jingxian Wu

[email protected]

This work is licensed under:

Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)

OUTLINE

• Chapter 1: Continuous-Time Signals ………………………. 3

• Chapter 2: Continuous-Time Systems ……………………… 45

• Chapter 3: Fourier Series ……………………………………. 84

• Chapter 4: Fourier Transform ……………………………… 122

• Chapter 5: Laplace Transform ……………………………… 170

• Chapter 6: Discrete-time Signals and Systems ……………… 222

2



Ch. 1 Continuous-Time Signals

Dr. Jingxian Wu

[email protected]

OUTLINE

4

• Introduction: what are signals and systems?

• Signals

• Classifications

• Basic Signal Operations

• Elementary Signals

INTRODUCTION

• Examples of signals and systems (Electrical Systems)

– Voltage divider

• Input signal: x = 5V

• Output signal: y = Vout

• The system output is a fraction of the input (𝑦 =𝑅2

𝑅1+𝑅2𝑥)

– Multimeter

• Input: the voltage across the battery

• Output: the voltage reading on the LCD display

• The system measures the voltage across two points

– Radio or cell phone

• Input: electromagnetic signals

• Output: audio signals

• The system receives electromagnetic signals and convert them to

audio signal

Voltage divider

multimeter

INTRODUCTION

• Examples of signals and systems (Biomedical Systems)

– Central nervous system (CNS)

• Input signal: a nerve at the finger tip senses the high

temperature, and sends a neural signal to the CNS

• Output signal: the CNS generates several output signals

to various muscles in the hand

• The system processes input neural signals, and generate

output neural signals based on the input

– Retina

• Input signal: light

• Output signal: neural signals

• Photosensitive cells called rods and cones in the retina convert

incident light energy into signals that are carried to the brain by the

optic nerve.

Retina

INTRODUCTION

• Examples of signals and systems (Biomedical Instrument)

– EEG (Electroencephalography) Sensors

• Input: brain signals

• Output: electrical signals

• Converts brain signal into electrical signals

– Magnetic Resonance Imaging (MRI)

• Input: when apply an oscillating magnetic field at a certain frequency,

the hydrogen atoms in the body will emit radio frequency signal,

which will be captured by the MRI machine

• Output: images of a certain part of the body

• Use strong magnetic fields and radio waves to form images of the

body.

MRI

EEG signal collection

INTRODUCTION

• Signals and Systems

– Even though the various signals and systems

could be quite different, they share some

common properties.

– In this course, we will study:

• How to represent signal and system?

• What are the properties of signals?

• What are the properties of systems?

• How to process signals with system?

– The theories can be applied to any general

signals and systems, be it electrical,

biomedical, mechanical, or economical, etc.

OUTLINE

9


• Signals

• Classifications



SIGNALS AND CLASSIFICATIONS

• What is signal?

– Physical quantities that carry information and changes with respect to time.

– E.g. voice, television picture, telegraph.

• Electrical signal

– Carry information with electrical parameters (e.g. voltage, current)

– All signals can be converted to electrical signals

• Speech →Microphone → Electrical Signal → Speaker → Speech

– Signals changes with respect to time

10

audio signal


• Mathematical representation of signal:

– Signals can be represented as a function of time t

– Support of signal:

– E.g.

– E.g.

• and are two different signals!

– The mathematical representation of signal contains two components:

• The expression:

• The support:

– The support can be skipped if

– E.g.

)2sin()(1 tts =

),(ts21 ttt

21 ttt

+− t

)2sin()(2 tts = t0

)(1 ts )(2 ts

)(ts

21 ttt

+− t

)2sin()(1 tts =

11


• Classification of signals: signals can be classified as

– Continuous-time signal v.s. discrete-time signal

– Analog signal v.s. digital signal

– Finite support v.s. infinite support

– Even signal v.s. odd signal

– Periodic signal v.s. Aperiodic signal

– Power signal v.s. Energy signal

– ……

12

OUTLINE

13


• Signals

• Classifications



SIGNALS: CONTINUOUS-TIME V.S. DISCRETE-TIME

• Continuous-time signal

– If the signal is defined over continuous-time, then the signal is a

continuous-time signal

• E.g. sinusoidal signal

• E.g. voice signal

• E.g. Rectangular pulse function

)4sin()( tts =

=otherwise,0

10,)(p

tAt

0 1 t

A

)(p t

14

Rectangular pulse function

• Discrete-time signal

– If the time t can only take discrete values, such as,

skTt = ,2,1,0 =k

then the signal is a discrete-time signal

– E.g. the monthly average precipitation at Fayetteville, AR (weather.com)

)()( skTsts =

– What is the value of s(t) at ?

• Discrete-time signals are undefined at !!!

• Usually represented as s(k)

ss kTtTk − )1(

month 1 =sT

skTt

12 , 2, 1, =k

SIGNALS: CONTINUOUS-TIME V.S. DISCRETE-TIME15

Monthly average precipitation

• Analog v.s. digital

– Continuous-time signal

• continuous-time, continuous amplitude→ analog signal

– Example: speech signal

• Continuous-time, discrete amplitude

– Example: traffic light

– Discrete-time signal

• Discrete-time, discrete-amplitude → digital signal

– Example: Telegraph, text, roll a dice

• Discrete-time, continuous-amplitude

– Example: samples of analog signal,

average monthly temperature

SIGNALS: ANALOG V.S. DIGITAL

10

2

3

0

21

10

23

0

21

16

Different types of signals

• Even v.s. odd

– x(t) is an even signal if:

• E.g.

– x(t) is an odd signal if:

• E.g.

– Some signals are neither even, nor odd

• E.g.

– Any signal can be decomposed as the sum of an even signal and odd

signal

• proof

SIGNALS: EVEN V.S. ODD

tetx =)(

)2cos()( ttx =

)2sin()( ttx =

0),2cos()( = tttx

)()( txtx −=

)()( txtx −=−

even odd

)()()( tytyty oe +=

17


• Example

– Find the even and odd decomposition of the following signal

tetx =)(

• Example

– Find the even and odd decomposition of the following signal


19

=otherwise0

0),4sin(2)(

tttx

• Periodic signal v.s. aperiodic signal

– An analog signal is periodic if

• There is a positive real value T such that

• It is defined for all possible values of t, (why?)

– Fundamental period : the smallest positive integer that satisfies

• E.g.

– So is a period of s(t), but it is not the fundamental period of

s(t)

SIGNALS: PERIODIC V.S. APERIODIC

)()( nTtsts +=

− t

0T

)()( 0nTtsts +=

01 2TT =

)()2()( 01 tsnTtsnTts =+=+

1T

20

0T

• Example

– Find the period of

– Amplitude: A

– Angular frequency:

– Initial phase:

– Period:

– Linear frequency:

)cos()( 0 += tAts − t

0

=0T

=0f


21


• Complex exponential signal

– Euler formula:

– Complex exponential signal

)sin()cos( xjxe jx +=

)sin()cos( 000 tjtetj

+=

– Complex exponential signal is periodic with period0

0

2

=

T

• Proof:

Complex exponential signal has same period as sinusoidal signal!

22

• The sum of two periodic signals is periodic if and only if the ratio of

the two periods can be expressed as a rational number.

• The period of the sum signal is


• The sum of two periodic signals

– x(t) has a period

– y(t) has a period

– Define z(t) = a x(t) + b y(t)

– Is z(t) periodic?

k

l

T

T=

2

1

2T

)()()( TtbyTtaxTtz +++=+

• In order to have x(t)=x(t+T), T must satisfy

• In order to have y(t)=y(t+T), T must satisfy

• Therefore, if

1kTT =

2lTT =

21 lTkTT ==)()()()()()( 21 tztbytaxlTtbykTtaxTtz =+=+++=+

1T

23

21 lTkTT ==

• Example

)3

sin()( ttx

= )9

2exp()( tjty

= )

9

2exp()( tjtz =

– Find the period of

– Is periodic? If periodic, what is the period?



)(),(),( tztytx

)(3)(2 tytx −

)()( tztx +

• Aperiodic signal: any signal that is not periodic.

)()( tzty


24

SINGALS: ENERGY V.S. POWER

• Signal energy

– Assume x(t) represents voltage across a resistor with resistance R.

– Current (Ohm’s law): i(t) = x(t)/R

– Instantaneous power:

– Signal power: the power of signal measured at R = 1 Ohm: )()( 2 txtp =

],[ ttt nn + )(tp

tnt

)( ntp

t

– Signal energy at:

ttpE nn )(

– Total energy

=→

n

nt

ttpE )(lim0

+

−= dttp )(

+

−= dttxE

2)(

– Review: integration over a signal gives the area under the signal.

Rtxtp /)()( 2=

25

Instantaneous power


• Energy of signal x(t) over

−= dttxE

2)(

• Average power of signal x(t)

−→=

T

TTdttx

TP

2)(

2

1lim

– If then x(t) is called an energy signal.,0 E

],[ +−t

– If then x(t) is called a power signal.,0 P

• A signal can be an energy signal, or a power signal, or neither, but not both.

26


• Example 1: )exp()( tAtx −=

• Example 2:

dttxT

PT

=0

2)(

1

0t

• All periodic signals are power signal with average power:

)sin()( 0 += tAtx

• Example 3: tjejtx )1()( += 100 t

27

OUTLINE

28


• Signals

• Classifications



OPERATIONS: SHIFTING

• Shifting operation

– : shift the signal x(t) to the right by )( 0ttx − 0t

– Why right?

Ax =)0( )()( 0ttxty −= Axttxty ==−= )0()()( 000

)()0( 0tyx =

29

Shifting to the right by two units

OPERATIONS: SHIFTING

• Example

o.w.

32

20

01

0

3

1

1

)(

−

+−

+

=t

t

t

t

t

tx

– Find )3( +tx

30

OPERATIONS: REFLECTION

• Reflection operation

– is obtained by reflecting x(t) w.r.t. the y-axis (t = 0))( tx −

31

-2 -1 1 2 3

-1

1

2

t

x(t)

-3 -2 -1 1

-1

1

2

t

x(-t)

Reflection

OPERATIONS: REFLECTION

• Example:

−+

=

o.w.

20

01

0

1

1

)( t

tt

tx

– Find x(3-t)

• The operations are always performed w.r.t. the time variable t directly!

32

OPERATIONS: TIME-SCALING

• Time-scaling operation

– is obtained by scaling the signal x(t) in time.

• , signal shrinks in time domain

• , signal expands in time domain

1a

1a

)(atx

33

-1 1

1

2

t

x(t)

-1.5 -1 -0.5 0.5 1 1.5

1

2

t

x(2t)

-2 -1 1 2

1

2

t

x(t/2)

Time scaling

OPERATIONS: TIME-SCALING

• Example:

o.w.

32

20

01

0

3

1

1

)(

−

+−

+

=t

t

t

t

t

tx

)( batx + 1. scale the signal by a: y(t) = x(at)

2. left shift the signal by b/a: z(t) = y(t+b/a) = x(a(t+b/a))=x(at+b)

• The operations are always performed w.r.t. the time variable t directly (be

careful about –t or at)!

)63( −tx– Find

34

OUTLINE

35

• Signals

• Classifications



ELEMENTARY SIGNALS: UNIT STEP FUNCTION

• Unit step function

0

0

,0

,1)(

=t

ttu

−

=

otherwise0,22

,1

)(t

tp

• Example: rectangular pulse

Express as a function of u(t) )(tp

36

1

1

t

u(t)

t

u(t)

Ã /2

1/ Ã

- Ã /2

Unit step function

Rectangular pulse

ELEMENTARY SIGNALS: RAMP FUNCTION

• The Ramp function

)()( tuttr =

t

)(tr

0

– The Ramp function is obtained by integrating the unit step function u(t)

= −dttu

t

)(

37

Unit ramp function

ELEMENTARY SIGNALS: UNIT IMPULSE FUNCTION

• Unit impulse function (Dirac delta function)

=

=

=

− 0,0

0,1)(

0,0)(

)0(

t

tdtt

tt

t

)(t

– delta function can be viewed as the limit of the rectangular pulse

)(lim)(0

tpt Δ→

=

– Relationship between and u(t)

dt

tdut

)()( =

0 t

)(t

)()( tudttt

= −

38

t

u(t)

Ã /2

1/ Ã

- Ã /2

Unit impulse function

Rectangular pulse


• Sampling property

+

−=− )()()( 00 txdttttx

)()()()( 000 tttxtttx −=−

• Shifting property

– Proof:

39


• Scaling property

+=+

a

bt

abat

||

1)(

– Proof:

40


• Examples

=−+− dtttt )3()(4

2

2

=−+− dtttt )3()(1

2

2

=−−− dttt )42()1exp(3

2

41

ELEMENTARY SIGNALS: SAMPLING FUNCTION

• Sampling function

x

xxSa

sin)( =

– Sampling function can be viewed as scaled version of sinc(x)

)(sin

)(Sinc xsax

xx

==

42

t

sa(t)

-4 -3 -2 -1 1 2 3 4

1

t

sinc(t)

Sampling function

Sinc function

ELEMENTARY SIGNALS: COMPLEX EXPONENTIAL

• Complex exponential

– Is it periodic?

• Example:

– Use Matlab to plot the real part of

tjretx

)( 0)(+

=

)]4()2([)( )21( −−+= +− tutuetx tj

43

SUMMARY

• Signals and Classifications

– Mathematical representation

– Continuous-time v.s. discrete-time

– Analog v.s. digital

– Odd v.s. even

– Periodic v.s. aperiodic

– Power v.s. energy


– Time shifting

– reflection

– Time scaling


– Unit step, unit impulse, ramp, sampling function, complex exponential

44

),(ts21 ttt



Ch. 2 Continuous-Time Systems

Dr. Jingxian Wu

[email protected]

46

OUTLINE

• Classifications of continuous-time system

• Linear time-invariant system (LTI)

• Properties of LTI system

• System described by differential equations

47

CLASSIFICATIONS: SYSTEM DEFINITION

• What is system?

– A system is a process that transforms input signals into output signals

• Accept an input

• Process the input

• Send an output (also called: the response of the system to input)

– System examples:

• Radio: input: electrical signals from air, output: music

• Robot: input: electrical control signals, output: motion or action

• Continuous-time system

– A system in which continuous-time input signals are transformed to

continuous-time output signals

• Discrete-time system

– A system in which discrete-time input signals are transformed to discrete-time

output signals.

continuous-time

System

)(tx )(tyDiscrete-time

System

)(nx )(ny

Continuous-time system discrete-time system

CLASSIFICATIONS: SYSTEM DEFINITION

• Classifications

– Linear v.s. non-linear

– Time-invariant v.s. time-varying

– Dynamic v.s. static (memory v.s. memoryless)

– Causal v.s. non-causal

– Invertible v.s. non-invertible

– Stable v.s. non-stable

48

49

CLASSIFICATIONS: LINEAR AND NON-LINEAR

• Linear system

– Let be the response of a system to an input

– Let be the response of a system to an input

– The system is linear if the superposition principle is satisfied:

• 1. the response to is

• 2. the response to is

)(1 ty )(1 tx

)(2 ty )(2 tx

)()( 21 txtx + )()( 21 tyty +

)()( 21 txtx +

)(1 ty

Linear

System

)()( 21 tyty +

)(1 tx

• Non-linear system

– If the superposition principle is not satisfied, then the system is a

non-linear system

Linear system

50


• Example: check if the following systems are linear

– System 1:

– System 3: inductor. Input: i(t), output v(t)

)](exp[)( txty =

– System 2: charge a capacitor. Input: i(t), output v(t)

−=

t

diC

tv )(1

)(

dt

tdiLtv

)()( =


• Example

– System 4:

– System 5:

– System 6:

51

|)(|)( txty =

)()( 2 txty =

CLASSIFICATIONS: LINEAR V.S. NON-LINEAR

• Example:

– Amplitude Modulation:

• Is it linear?

52

Amplitude modulation

53CLASSIFICATIONS: TIME-VARYING V.S. TIME-INVARIANT

• Time-invariant

– A system is time-invariant if a time shift in the input signal causes an

identical time shift in the output signal

Time-invariant

System

)( 0ttx −)(ty Time-invariant

System

)( 0tty −)(tx

• Examples

– y(t) = cos(x(t))

– =t

dvvxty0

)()(

Time-invariant system

54

CLASSIFICATIONS: MEMORY V.S. MEMORYLESS

• Memoryless system

– If the present value of the output depends only on the present value of input, then the system is said to be memoryless (or instantaneous).

– Example: input x(t): the current passing through a resistor

output y(t): the voltage across the resistor

)()( tRxty =

– The output value at time t depends only on input value at time t.

• System with memory

– If the present value of the output depends on not only present value of input, but also previous input values, then the system has memory.

– Example: capacitor, current: x(t), output voltage: y(t)

=t

dxC

ty0

)(1

)(

– the output value at t depends on all input values before t

55

CLASSIFICATIONS: MEMORY V.S. MEMORYLESS

• Examples: determine if the systems has memory or not

– =

−=N

i

ii Ttxaty0

)()(

– )())(2sin()( 2 txtxty +=

56

CLASSIFICATIONS: CAUSAL V.S. NON-CAUSAL

• Causal system

– A system is causal if the output depends only on values of input

for

• The output depends on only input from the past and present

– Example

)( 0ty

0tt

)()( atxty +=

• Non-causal system

– A system is non-causal if the output depends on the input from the future (prediction).

– Examples:

0a

– The output value at t depends on the input value at t + a (from future)

=t

dxC

ty0

)(1

)(

– All practical systems are causal.

−=2/

2/)(

1)(

T

Tdx

Tty

57

CLASSIFICATION: INVERTIBILITY

• Invertible

– A system is invertible if

• by observing the output, we can determine its input.

SystemInverse

System

)(tx )(ty )(tx

– Question: for a system, if two different inputs result in the same

output, is this system invertible?• Example

)(2)( txty =

)(cos)( txty =

– If two different inputs result in the same output, the system is non-

invertible

invertible system

58

CLASSIFICATION: STABILITY

• Bounded signal

– Definition: a signal x(t) is said to be bounded if

Btx |)(|

• Bounded-input bounded-output (BIBO) stable system

– Definition: a system is BIBO stable if, for any bounded input x(t),

the response y(t) is also bounded.

21 |y(t)| |)(| BBtx

• Example: determine if the systems are BIBO stable

)(exp)( txty =

−=

t

dxty )()(

t

t

59

OUTLINE





60

LTI: DEFINTION

• Linear time-invariant (LTI) system

– Definition: a system is said to be LTI if it’s linear and time-invariant

)(txi

System)(tyi

– Linear

Input:

Output: =

=+++=N

i

iiNN tyatyatyatyaty1

2211 )()()()()(

=

=+++=N

i

iiNN txatxatxatxatx1

2211 )()()()()(

– Time-invariant

Input: )()( 0ttxtx i −=

Output: )()( 0ttyty i −=

system

61

LTI: IMPULSE RESPONSE

• Impulse response of LTI system

– Def: the output (response) of a system when the input is a unit impulse

function (delta function).

• Usually denoted as h(t)

• For system with an arbitrary input x(t), we want to find

out the output y(t).

– Method 1: differential equations

– Methods 2: convolution integral

– Methods 3: Laplace transform, Fourier transform,

)()( ttx =System

)()( thty =

LTI system

62

LTI: CONVOLUTION

• Derivation

– Any signal can be approximated by the sum of a sequence of delta

functions

+

−=→

+

−−=−=

n

ntnxdtxtx )()(lim)()()(0

+

−=→

+

−=

n

nzdz )(lim)(0

t

x(t)

integration

63

LTI: CONVOLUTION

• Derivation

– Any signal can be approximated by the sum of a sequence of delta

functions

+

−=→

+

−−=−=

n


)(tSystem

)(th

– Time invariant

)( −ntSystem

)( −nth

– Linear

−+

−=

)()( ntnxn

System

−+

−=

)()( nthnxn

LTI system

64

LTI: CONVOLUTION

• Convolution

)(txSystem

+

−−= dthxty )()()(

– Definition: the convolution of two signals x(t) and h(t) is defined as

+

−−= dthxty )()()(

– The operation of convolution is usually denoted with the symbol

+

−−== dthxthtxty )()()()()(

)(txh(t)

)()( thtx

For LTI system, if we know input x(t) and impulse response h(t),

Then the output is )()( thtx

LTI system

LTI system

65

LTI: CONVOLUTION

• Examples

)()( ttx

)()( tutx

)()( 0tttx −

66

LTI: CONVOLUTION

• Examples

)()exp( tubt−)()exp( tuat−

?)( =ty

LTI system

LTI: CONVOLUTION

• Example

– Obtain the impulse response of a capacitor and use it to find the unit-step

response by using convolution. Assume the input is the current, and the

output is the voltage. Let C = 1F.

67

−=

t

diC

tv )(1

)(

68

LTI: CONVOLUTION PROPERTIES

• Commutativity

)()()()( txtytytx =

– Proof:

+

−−= dtyxtytx )()()()(

)(txh(t)

)()( thtx )(thx(t)

)()( txth ➔

commutativity

69


• Associativity

)()()()()()()()()( 212121 ththtxththtxththtx ==

– proof

)(tx)()( 21 thth

)(ty)(tx)(1 th )(1 th )(2 th

)(ty➔

)(1 ty

)(th

Associativity

70


• Distributivity

)()()()()()()( 1121 thtxthtxththtx +=+

– proof

)(tx

)(1 th

)(2 th

)(ty

+)(tx

)()( 21 thth +)(ty

➔

Distributivity

71


• Example

)(tx

)(1 th

)(3 th

)(ty

+

)(2 th

)(4 th

)()2exp()(1 tutth −= )()exp(2)(2 tutth −=)()3exp()(3 tutth −= )(4)(4 tth =

?)( =th

72

LTI: GRAPHICAL CONVOLUTION

• Graphical interpretation of convolution

– 1. Reflection )()( −= hg )())(()( 000 −=−−=− ththtg

– 3. Multiplication )()( 0 −thx

+

−−= dthxty )()()(

– 4. Integration +

−−= dthxty )()()( 00

)())(()( 000 −=−−=− ththtg

)(x )(h

t

x(t)

t

x(t)

t

x(t)

t

h(-t)

t

73

LTI: GRAPHICAL CONVOLUTION

• Example

)](2[)](2[)( 22 atpatpaty aa −=

74

OUTLINE





75

LTI PROPERTIES

• Memoryless LTI system

– Review: present output only depends on present input

)()( tKxty =

0for t

– The impulse response of Memoryless LTI system is

• Causal LTI system

– Review: output depends on only current input and past input.

– The impulse response of causal LTI system must satisfy:

)()( tKth =

0)( =th

– Why?

76

LTI PROPERTIES

• Invertible LTI Systems

– Review: a system is invertible iff (if and only if) there is an inverse system that, when connected in cascade with the original system, yields an output equal to original system input

h(t) g(t))(tx )(ty )(tx

)()()()( txtgthtx =

– For invertible LTI systems with IR (impulse response) , there exists inverse system such that

)(th)(tg

)()()( tthtg =

– Example: find the inverse system of LTI system )()( 0ttth −=

77

LTI PROPERTIES

• BIBO Stable LTI state

– Review: a system is BIBO stable iff every bounded input produces a

bounded output.

– LTI system: an LTI system is BIBO stable iff

+

−dtth )(

• Proof:

78

LTI PROPERTIES

• Examples

– Determine: causal or non-causal, memory or memoryless, stable or unstable

– 1.

– 2.

– 3.

)1()()3exp()()2exp()(1 −+−+−= ttuttuttth

)()2exp(3)(2 tutth −=

)5(5)(3 += tth

79

OUTLINE





• LTI system can be represented by differential equations

– Initial conditions:

– Notation: n-th derivative:

DIFFERENTIAL EQUATIONS

80

)()(')()()(')( )(

10

)(

10 txbtxbtxbtyatyatya M

M

N

N +++=+++

n

nn

dt

tydty

)()()( =

0

)(

=t

k

k

dt

tyd1,,0 −= Nk

81

DIFFERENTIAL EQUATION

• Example:

– Consider a circuit with a resistor R = 1 Ohm and an inductor L = 1H, with

a voltage source v(t) = Bu(t), and is the initial current in the inductor.

The output of the system is the current across the inductor.

• Represent the system with a differential equation.

• Find the output of the system with and

oI

0=oI 1=oI


• Zero-state response

– The output of the system when the initial conditions are zero

– Denoted as

• Zero-input response

– The output of the system when the input is zero

– Denoted as

• The actual output of the system

82

)()(')()()(')( )(

10

)(

10 txbtxbtxbtyatyatya M

M

N

N +++=+++

0

)(

=t

k

k

dt

tyd1,,0 −= Nk

)(tyzs

)(tyzi

)()()( tytyty zizs +=


• Example

– Find the zero-state output and zero-input response of the RL circuit in the

previous example.

83



Ch. 3 Fourier Series

Dr. Jingxian Wu

[email protected]

85

OUTLINE

• Introduction

• Fourier series

• Properties of Fourier series

• Systems with periodic inputs

86

INTRODUCTION: MOTIVATION

• Motivation of Fourier series

– Convolution is derived by decomposing the signal into the sum of a series

of delta functions

• Each delta function has its unique delay in time domain.

• Time domain decomposition

+

−=→

+

−−=−=

n


t

x(t)

Illustration of integration


• Can we decompose the signal into the sum of other functions

– Such that the calculation can be simplified?

– Yes. We can decompose periodic signal as the sum of a sequence of

complex exponential signals ➔ Fourier series.

– Why complex exponential signal? (what makes complex exponential

signal so special?)

• 1. Each complex exponential signal has a unique frequency ➔

frequency decomposition

• 2. Complex exponential signals are periodic

87

tfjtjee 00 2

=

2

00

=f

Department of Engineering Science

Sonoma State University

88

INTRODUCTION: REVIEW

• Complex exponential signal

)2sin()2cos(2 ftjfte ftj +=

– Complex exponential function has a one-to-one relationship with

sinusoidal functions.

– Each sinusoidal function has a unique frequency: f

• What is frequency?

– Frequency is a measure of how fast the signal can change within a

unit time.

• Higher frequency ➔ signal changes faster

f = 0 Hz

f = 1 Hz

f = 3 Hz

Sinusoidal at different frequencies

89

INTRODUCTION: ORTHONORMAL SIGNAL SET

• Definition: orthogonal signal set

– A set of signals, , are said to be orthogonal over an

interval (a, b) if ),(),(),( 210 ttt

kl

klCdttt

b

akl

=

= ,0

,)()( *

• Example:

– the signal set: are

orthogonal over the interval , where

tjk

k et 0)(

= ,2,1,0 =k],0[ 0T

0

0

2

T

=

90

OUTLINE

• Introduction

• Fourier series



91

FOURIER SERIES

• Definition:

– For any periodic signal with fundamental period , it can be decomposed as the sum of a set of complex exponential signals as

tjn

n

nectx 0)(

+

−=

=

• , Fourier series coefficients,2,1,0, =ncn

−=

0

0)(1

0T

tjn

n dtetxT

c

• derivation of :nc

0T

00

2

T

=

For a periodic signal, it can be either represented as s(t), or represented as

92

FOURIER SERIES

• Fourier series

tjn

n

nectx 0)(

+

−=

=

– The periodic signal is decomposed into the weighted summation of a set of orthogonal complex exponential functions.

– The frequency of the n-th complex exponential function:

,2,1,0, =ncn

nc

0n

• The periods of the n-th complex exponential function:

– The values of coefficients, , depend on x(t)

• Different x(t) will result in different

• There is a one-to-one relationship between x(t) and

n

TTn

0=

nc

)(ts ],,,,,[ 210,12 ccccc −−➔

nc

93

FOURIER SERIES

• Example

10

01

,

,)(

−

−

=t

t

K

Ktx

-3 -2 -1 1 2

t

x(t)

Rectangle pulses

FOURIER SERIES

• Amplitude and phase

– The Fourier series coefficients are usually complex numbers

– Amplitude line spectrum: amplitude as a function of

– Phase line spectrum: phase as a function of

94

nnn jbac +=

22

nnn bac +=

n

nn

a

btana=

0n

0n

nj

n ec

=

95

FOURIER SERIES: FREQUENCY DOMAIN

• Signal represented in frequency domain: line spectrum

– Each has its own frequency

– The signal is decomposed in frequency domain.

– is called the harmonic of signal s(t) at frequency

– Each signal has many frequency components.

• The power of the harmonics at different frequencies determines

how fast the signal can change.

nc

nc

amplitude phase

0n

0n

FOURIER SERIES: FREQUENCY DOMAIN

• Example: Piano Note

96

E5: 659.25 Hz

E6: 1318.51 Hz

B6: 1975.53 Hz

E7: 2637.02 Hz

E5

E6B6

E7

All graphs in this page are created by using the open-source software Audacity.

piano notes

One piano note

spectrum

97

FOURIER SERIES

• Example

– Find the Fourier series of )exp()( 0tjts =

FOURIER SERIES

• Example

– Find the Fourier series of

98

)cos()( 0 ++= tABts

)100sin(1)( tty +=

Time domain Amplitude spectrum Phase spectrum

99

FOURIER SERIES

• Example

– Find the Fourier series of

−

−−

=

2/2/,0

2/2/,

2/2/,0

)(

Tt

tK

tT

ts

5,1 == T

10,1 == T

15,1 == T

)(csinT

n

T

Kcn

=

t

x(t)

Time domain

100

FOURIER SERIES: DIRICHLET CONDITIONS

• Can any periodic signal be decomposed into Fourier series?

– Only signals satisfy Dirichlet conditions have Fourier series

• Dirichlet conditions

– 1. x(t) is absolutely integrable within one period

Tdttx |)(|

– 2. x(t) has only a finite number of maxima and minima.

– 3. The number of discontinuities in x(t) must be finite.

101

OUTLINE

• Introduction

• Fourier series



102

PROPERTIES: LINEARITY

• Linearity

– Two periodic signals with the same period

0

0

2

=

T

– The Fourier series of the superposition of two signals is

+

−=

+=+

n

tjn

nn ekktyktxk 0)()()( 2121

+

−=

=

n

tjn

netx 0)(

)()()( 2121 nn kktyktxk +=+

– If

ntx =)(nty =)(

• then

+

−=

=

n

tjn

nety 0)(

103

PROPERTIES: EFFECTS OF SYMMETRY

• Symmetric signals

– A signal is even symmetry if:

– A signal is odd symmetry if:

– The existence of symmetries simplifies the computation of Fourier series

coefficients.

)()( txtx −=

)()( txtx −−=

-4 -3 -2 -1 1 2 3 4

t

x(t)

-5 -4 -3 -2 -1 1 2 3 4 5

t

x(t)

Even symmetric Odd symmetric

104


• Fourier series of even symmetry signals

– If a signal is even symmetry, then

( )+

−=

=n

n tnatx 0cos)( ( ) =2/

00

0

0

cos)(2 T

n dttntxT

a

• Fourier series of odd symmetry signals

– If a signal is odd symmetry, then

( )+

=

=1

0sin)(n

n tnbtx ( ) =2/

00

0

0

sin)(2 T

n dttntxT

b

105


• Example

−

−=

TtTAtT

A

TttT

AA

tx

2/,34

2/0,4

)(t

x(t)

Graph of x(t)

106

PROPERTIES: SHIFT IN TIME

• Shift in time

– If has Fourier series , then has Fourier series )(tx nc )( 0ttx −

00tjn

nec−

)(tx nc➔if , then )( 0ttx − ➔00tjn

nec−

– Proof:

107

PROPERTIES: PARSEVAL’S THEOREM

• Review: power of periodic signal

=T

dttxT

P0

2|)(|1

• Parseval’s theorem

+

−=

=m

m

T

dttxT

2

0

2 |||)(|1

)(txif ➔ n

then

– Proof:

The power of signal can be computed in frequency domain!

108

PROPERTIES: PARSEVAL’S THEOREM

• Example

– Use Parseval’s theorem find the power of )sin()( 0tAtx =

109

OUTLINE

• Introduction

• Fourier series



110

PERIODIC INPUTS: COMPLEX EXPONENTIAL INPUT

• LTI system with complex exponential input

tjetx =)()(th

)(ty

)()()()()( txththtxty ==

dtxh+

−−= )()(

djhtj +

−−= )exp()()exp(

djhH +

−−= )exp()()(

• Transfer function

– For LTI system with complex exponential input, the output is

)exp()()( tjHty =

– It tells us the system response at different frequencies

PERIODIC INPUT

• Example:

– For a system with impulse response

find the transfer function

111

)()( 0ttth −=

112

PERIODIC INPUT:

• Example

– Find the transfer function of the system shown in figure.

RL circuit

PERIODIC INPUTS

• Example

– Find the transfer function of the system shown in figure

113

RC circuit

114

PERIODIC INPUTS: TRANSFER FUNCTION

• Transfer function

– For system described by differential equations

= =

=n

i

m

i

i

i

i

i txqtyp0 0

)()( )()(

=

=

=n

i

i

i

m

i

i

i

jp

jq

H

0

0

)(

)(

)(

115

PERIODIC INPUTS

• LTI system with periodic inputs

– Periodic inputs:

tjne 0

)(th)( 0

0

nHetjn

+

−=

=n

n tjnctx )exp()( 0

linear: tjn

n

nec 0+

−=

)(th

)( 00

+

−=

nHectjn

n

n

)(tx)(th

)( 00

+

−=

nHectjn

n

n

For system with periodic inputs, the output is the weighted

sum of the transfer function.

T

20 =

116

PERIODIC INPUTS

• Procedures:

– To find the output of LTI system with periodic input

• 1. Find the Fourier series coefficients of periodic input x(t).

−

=T

tjn

n dtetxT 0

0)(1

• 2. Find the transfer function of LTI system

Tf

22 00 ==

period of x(t)

• 3. The output of the system is

)()( 00 =

+

−=

nHectytjn

n

n

)(H

117

PERIODIC INPUTS

• Example

– Find the response of the system when the input is

)2cos(2)cos(4)( tttx −=

RL Circuit

118

PERIODIC INPUTS

• Example

– Find the response of the system when the input is shown in figure.

-3 -2 -1 1 2

t

x(t)

RC circuitSquare pulses

PERIODIC INPUTS: GIBBS PHENOMENON

• The Gibbs Phenomenon

– Most Fourier series has infinite number of elements→ unlimited

bandwidth

• What if we truncate the infinite series to finite number of elements?

– The truncated signal, , is an approximation of the original

signal x(t)

119

tjn

n

nectx 0)(

+

−=

=

tjnN

Nn

nN ectx 0)(

+

−=

=

)(txN

PERIODIC INPUTS: GIBBS PHENOMENON

120

=

even. 0,

odd, ,12

n

nnj

K

cn tjn

N

Nn

nN ectx 0)(

+

−=

=

)(3 tx )(5 tx )(19 tx

-3 -2 -1 1 2

t

x(t)

Square pulses

FOURIER SERIES

• Analogy: Optical Prism

– Each color is an Electromagnetic wave with a different frequency

121

Optical prism



Ch. 4 Fourier Transform

Dr. Jingxian Wu

[email protected]

123

OUTLINE

• Introduction

• Fourier Transform

• Properties of Fourier Transform

• Applications of Fourier Transform

124


• Motivation:

– Fourier series: periodic signals can be decomposed as the summation of

orthogonal complex exponential signals

tjnctxn

n 0exp)( +

−=

=

• each harmonic contains a unique frequency: n/T

=T

n dttjntxT

c0

0exp)(1

How about aperiodic signals ?

( )=T• time domain ➔ frequency domain

t

x(t)

Time domain Frequency domain

125

INTRODUCTION: TRANSFER FUNCTION

• System transfer function

• System with periodic inputs

tje

)(th)( He tj

+

−= dttjthH exp)()(

tjne 0

)(th)( 0

0 nHe

tjn

tjn

n

nec 0+

−= )(th)( 0

0 nHec

tjn

n

n+

−=

)(tx)(th

)( 00

nHectjn

n

n+

−=

126

OUTLINE

• Introduction




127

FOURIER TRANSFORM

• Inverse Fourier Transform


– given x(t), we can find its Fourier transform

– given , we can find the time domain signal x(t)

– signal is decomposed into the “weighted summation” of complex exponential functions. (integration is the extreme case of summation)

+

−

−= dtetxX tj )()(

+

−=

deXtx tj)(2

1)(

)(X

)(X

➔)(tx )(X

128

FOURIER TRANSFORM

• Example

– Find the Fourier transform of )/()( trecttx =

t

x(t)

t

x(t)

129

FOURIER TRANSFORM

• Example

– Find the Fourier transform of |)|exp()( tatx −= 0a

130

FOURIER TRANSFORM

• Example

– Find the Fourier transform of 0a)()exp()( tuattx −=

131

FOURIER TRANSFORM

• Example

– Find the Fourier transform of )()( attx −=

132

FOURIER TRANSFORM: TABLE

133

FOURIER TRANSFORM

+

−dttx |)(|

)()exp()( tuttx =

• Example

–

• The existence of Fourier transform

– Not all signals have Fourier transform

– If a signal have Fourier transform, it must satisfy the following two

conditions

• 1. x(t) is absolutely integrable

• 2. x(t) is well behaved

– The signal has finite number of discontinuities, minima,

and maxima within any finite interval of time.

134

OUTLINE

• Introduction




135


• Linearity

– If

– then

)()( 11 Xtx )()( 22 Xtx

)()()()( 2121 bXaXtbxtax ++

• Example

– Find the Fourier transform of )(4)()2exp(3)/(2)( ttuttrecttx +−+=

136

PROPERTY: TIME-SHIFT

• Time shift

– If

– Then)()( Xtx

]exp[)()( 00 tjXttx −−

• Review: complex number

jbacjcecc j +=+== )sin(||)cos(||||

cos|| ca = sin|| cb =

22|| bac += )/tan( aba=

phase shift

time shift in time domain ➔ frequency shift in frequency domain

– Phase shift of a complex number c by : 0 )(exp||)exp( 00 += jcjc

137

PROPERTY: TIME SHIFT

• Example:

– Find the Fourier transform of 2)( −= trecttx

138

PROPERTY: TIME SCALING

• Time scaling

– If

– Then

• Example

– Let , find the Fourier transform of

)()( Xtx

aX

aatx

||

1)(

( ) 2/1)( −= rectX )42( +− tx

139

PROPERTY: SYMMETRY

• Symmetry

– If , and is a real-valued time signal

– Then

)()( Xtx )(tx

)()( * XX =−

140

PROPERTY: DIFFERENTIATION

• Differentiation

– If

– Then

)()( Xtx

)()(

Xjdt

tdx ( ) )(

)( Xj

dt

txd n

n

n

• Example

– Let , find the Fourier transform of ( ) 2/1)( −= rectXdt

tdx )(

141

PROPERTY: DIFFERENTIATION

• Example

– Find the Fourier transform of

(Hint: )

)sgn()( ttx =

)()sgn(2

1tt

dt

d=

142

PROPERTY: CONVOLUTION

• Convolution

– If ,

– Then

)()( Xtx )()( Hth

)()()()( HXthtx

)(tx

)(th)()( thtx )(X

)(H)()( HX

time domain frequency domain

143

PROPERTY: CONVOLUTION

• Example

– An LTI system has impulse response

If the input is

Find the output

( ) )(exp)( tuatth −=

( ) )()exp()()(exp)()( tuctactubtbatx −−+−−=

)0,0,0( cba

144

PROPERTY: MULTIPLICATION

• Multiplication

– If ,

– Then

)()( Xtx )()( Mtm

)()(2

1)()(

MXtmtx

145

PROPERTY: DUALITY

• Duality

– If

– Then

)()( Gtg

)(2)( − gtG

146

PROPERTY: DUALITY

• Example


(recall: )

=

2)(

tSath

2sinc )/(rect t

147

PROPERTY: DUALITY

• Example

– Find the Fourier transform of 1)( =tx

tjetx 0)(

=– Find the Fourier transform of

148

PROPERTY: SUMMARY

149

PROPERTY: EXAMPLES

• Examples

– 1. Find the Fourier transform of )cos()( 0ttx =

– 2. Find the Fourier transform of )()( tutx =

1)sgn(2

1)( += ttu

jt

2)sgn(

150

PROPERTY: EXAMPLES

• Examples

– 3. A LTI system with impulse response

Find the output when input is )(exp)( tuatth −=

)()( tutx =

– 4. If , find the Fourier transform of

(Hint: )

)()( Xtx −

t

dx )(

)()()( tutxdxt

= −

151

PROPERTY: EXAMPLES

• Example

– 5. (Modulation) If ,

Find the Fourier transform of )()( Xtx )cos()( 0ttm =

)()( tmtx

– 6. If , find x(t)

ja

X+

=1

)(

152

PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN

• Differentiation in frequency domain

– If:

– Then:

)()( Xtx

n

nn

d

Xdtxjt

)()()( =−

PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN153

),()exp( tuatt − 0a

• Example


154

PROPERTY: FREQUENCY SHIFT

• Frequency shift

– If:

– Then:

)()( Xtx

)()exp()( 00 − Xtjtx

• Example

– If , find the Fourier transform ( ) 2/1)( −= rectX )2exp()( tjtx −

155

PROPERTY: PARSAVAL’S THEOREM

• Review: signal energy

+

−= dttxE 2|)(|

• Parsaval’s theorem

+

−

+

−=

dXdttx 22 |)(|

2

1|)(|

156

PROPERTY: PARSAVAL’S THEOREM

• Example:

– Find the energy of the signal )()2exp()( tuttx −=

157

PROPERTY: PERIODIC SIGNAL

• Fourier transform of periodic signal

– Periodic signal can be written as Fourier series

tjnctxn

n 0exp)( +

−=

=

– Perform Fourier transform on both sides

)(2)( 0 ncXn

n −= +

−=

158

OUTLINE

• Introduction




159

APPLICATIONS: FILTERING

• Filtering

– Filtering is the process by which the essential and useful part of a signal is

separated from undesirable components.

• Passing a signal through a filter (system).

• At the output of the filter, some undesired part of the signal (e.g. noise)

is removed.

– Based on the convolution property, we can design filter that only allow

signal within a certain frequency range to pass through.

)(tx

)(th)()( thtx )(X

)(H)()( HX

time domain frequency domain

filter filter

160


• Classifications of filters

Low pass filter

Band pass filterBand stop (Notch) filter

PassbandStop

band PassbandStop

band

High pass filter

Passband Stop

band

Stop

band

Stop

bandPassband Passband

161

APPLICATION: FILTERING

• A filtering example

– A demo of a notch filter

)(X

)(H

)()( HX

Corrupted sound Filter Filtered sound

162


• Example

– Find out the frequency response of the RC circuit

– What kind of filters it is?

RC circuit

163

APPLICATION: SAMPLING THEOREM

• Sampling theorem: time domain

– Sampling: convert the continuous-time signal to discrete-time signal.

+

−=

−=n

nTttp )()(

sampling period

)()()( tptxtxs =

)(tx

Sampled signal

164


• Sampling theorem: frequency domain

– Fourier transform of the impulse train

• impulse train is periodic

+

−=

+

−=

=−=n

tjn

sn

sse

TnTttp

11

)()(

• Find Fourier transform on both sides

+

−=

−=n

s

s

nT

P )(2

)(

• Time domain multiplication ➔ Frequency domain convolution

)()(2

1)()(

PXtptx

+

−=

−n

s

s

nXT

tptx )(1

)()(

s

sT

2=

Fourier series

165


• Sampling theorem: frequency domain

– Sampling in time domain ➔ Repetition in frequency domain

Time domain Frequency domain

166


• Sampling theorem

– If the sampling rate is twice of the bandwidth, then the original signal can

be perfectly reconstructed from the samples.

Bs 2

Bs 2

Bs 2=

Bs 2

Frequency domain

167

APPLICATION: AMPLITUDE MODULATION

• What is modulation?

– The process by which some characteristic of a carrier wave is

varied in accordance with an information-bearing signal

modulationInformation

bearing signal

Carrier wave

Modulated signal

• Three signals:

– Information bearing signal (modulating signal)

• Usually at low frequency (baseband)

• E.g. speech signal: 20Hz – 20KHz

– Carrier wave

• Usually a high frequency sinusoidal (passband)

• E.g. AM radio station (1050KHz) FM radio station

(100.1MHz), 2.4GHz, etc.

– Modulated signal: passband signal

168


• Amplitude Modulation (AM)

)2cos()()( tftmAts cc =

– A direct product between message signal and carrier signal

Mixer

Local

Oscillator

)(tm

)2cos( tfA cc

)(ts


169


• Amplitude Modulation (AM)

)()(2

)( ccc ffMffM

AfS ++−=




Ch. 5 Laplace Transform

Dr. Jingxian Wu

[email protected]

171

OUTLINE

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Laplace Transform

• Applications of Laplace Transform

172

INTRODUCTION

• Why Laplace transform?

– Frequency domain analysis with Fourier transform is extremely useful for

the studies of signals and LTI system.

• Convolution in time domain ➔Multiplication in frequency domain.

– Problem: many signals do not have Fourier transform

0),()exp()( = atuattx )()( ttutx =

– Laplace transform can solve this problem

• It exists for most common signals.

• Follow similar property to Fourier transform

• It doesn’t have any physical meaning; just a mathematical tool

to facilitate analysis.

– Fourier transform gives us the frequency domain

representation of signal.

173

OUTLINE

• Introduction



• Inverse Lapalace Transform


174LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM

• Bilateral Laplace transform (two-sided Laplace transform)

,)exp()()( +

−−= dtsttxsX B

– is a complex variable

– s is often called the complex frequency

– Notations:

– : a function of time t → x(t) is called the time domain signal

– a function of s → is called the s-domain signal

– S-domain is also called as the complex frequency domain

js +=

)()( sXtx B

js +=

)]([)( txLsX B =

)(tx

:)(sX B)(sX B

• Time domain v.s. S-domain

LAPLACE TRANSFORM

• Time domain v.s. s-domain

– : a function of time t → x(t) is called the time domain signal

– a function of s → is called the s-domain signal

• S-domain is also called the complex frequency domain

– By converting the time domain signal into the s-domain, we can usually

greatly simplify the analysis of the LTI system.

– S-domain system analysis:

• 1. Convert the time domain signals to the s-domain with the Laplace

transform

• 2. Perform system analysis in the s-domain

• 3. Convert the s-domain results back to the time-domain

175

)(tx:)(sX B

)(sX B

176

• Example

– Find the Bilateral Laplace transform of )()exp()( tuattx −=

• Region of Convergence (ROC)

– The range of s that the Laplace transform of a signal converges.

– The Laplace transform always contains two components

• The mathematical expression of Laplace transform

• ROC.

LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM

177

• Example

– Find the Laplace transform of )()exp()( tuattx −−−=


178

• Example

– Find the Laplace transform of )()exp(4)()2exp(3)( tuttuttx −+−=


179LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

• Unilateral Laplace transform (one-sided Laplace transform)

+

−−=

0)exp()()( dtsttxsX

– :The value of x(t) at t = 0 is considered.

– Useful when we dealing with causal signals or causal systems.

• Causal signal: x(t) = 0, t < 0.

• Causal system: h(t) = 0, t < 0.

– We are going to simply call unilateral Laplace transform as

Laplace transform.

−0

+

−−=

0)exp()()( dtsttxsX

180

• Example: find the unilateral Laplace transform of the following

signals.

– 1. Atx =)(

– 2. )()( ttx =

LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

181

• Example

– 3. )2exp()( tjtx =

– 4.

)2sin()( ttx =– 5.

)2cos()( ttx =

LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM182

183

OUTLINE

• Introduction





184


• Linearity

– If

– Then

The ROC is the intersection between the two original signals

)()( 11 sXtx )()( 22 sXtx

)()()()( 2121 sbXsaXtbxtax ++

• Example

– Find the Laplace transfrom of )()exp( tubtBA −+

185

PROPERTIES: TIME SHIFTING

• Time shifting

– If and

– Then

The ROC remain unchanged

)()( sXtx

)exp()()()( 000 stsXttuttx −−−

00 t

186

PROPERTIES: SHIFTING IN THE s DOMAIN

• Shifting in the s domain

– If

– Then )()exp()()( 00 ssXtstxty −=

• Example

– Find the Laplace transform of )()cos()exp()( 0 tutatAtx −=

)Re(s)()( sXtx

)Re()Re( 0ss +

187

PROPERTIES: TIME SCALING

• Time scaling

– If

– Then )()( sXtx

1}Re{ as

a

sX

aatx

1)(

1}Re{ s

• Example

– Find the Laplace transform of )()( atutx =

188

PROPERTIES: DIFFERENTIATION IN TIME DOMAIN

• Differentiation in time domain

– If

– Then)()( sGtg

)0()()( −− gssG

dt

tdg

• Example

– Find the Laplace transform of ),(sin)( 2 tuttg =

)0()0()0()()( )1()2(1 −−−−−− −−−− nnnn

n

n

gsggssGsdt

tgd

0)0( =−g

)0(')0()()( 2

2

2−− −− gsgsGs

dt

tgd

189

PROPERTIES: DIFFERENTIATION IN TIME DOMAIN

• Example

– Use Laplace transform to solve the differential equation

,0)(2)('3)('' =++ tytyty 3)0( =−y 1)0(' =−y

190

PROPERTIES: DIFFERENTIATION IN S DOMAIN

• Differentiation in s domain

– If

– Then

)()( sXtx

n

nn

ds

sXdtxt

)()()( −

• Example

– Find the Laplace transform of )(tut n

191

PROPERTIES: CONVOLUTION

• Convolution

– If

– Then

The ROC of is the intersection of the ROCs of X(s) and

H(s)

)()( sXtx )()( sHth

)()()()( sHsXthtx

)()( sHsX

192

PROPERTIES: INTEGRATION IN TIME DOMAIN

• Integration in time domain

– If

– Then

)()( sXtx

)(1

)(0

sXs

dxt

• Example

– Find the Laplace transform of )()( ttutr =

193


• Example

– Find the convolution

−

−

a

atrect

a

atrect

22

194


• Example

– For a LTI system, the input is , and the output of the

system is )()2exp()( tuttx −=

)()3exp()2exp()exp()( tutttty −−−+−=

Find the impulse response of the system

195


• Example

– Find the Laplace transform of the impulse response of the LTI system

described by the following differential equation

)()('3)()('3)(''2 txtxtytyty +=+−

assume the system was initially relaxed ( )0)0()0( )()( == nn xy

196

PROPERTIES: MODULATION

• Modulation

– If

– Then

)()( sXtx )()( sXtx

)()(2

1)cos()( 000 jsXjsXttx −++

)()(2

)sin()( 000 jsXjsXj

ttx −−+

197

PROPERTIES: MODULATION

• Example

– Find the Laplace transform of )()sin()exp()( 0 tutattx −=

198

PROPERTIES: INITIAL VALUE THEOREM

• Initial value theorem

– If the signal is infinitely differentiable on an interval around

then

)(tx )0( +x

)(lim)0( ssXxs →

+ =

– The behavior of x(t) for small t is determined by the behavior of X(s) for large s.

=s must be in ROC

199

PROPERTIES: INITIAL VALUE THEOREM

• Example

– The Laplace transform of x(t) is

Find the value of ))(()(

bsas

dcssX

−−

+=

)0( +x

200

PROPERTIES: FINAL VALUE THEOREM

• Final value theorem

– If

– Then: )()( sXtx

)(lim)(lim0

ssXtxst →→

• Example

– The input is applied to a system with transfer

function , find the value of

0=s must be in ROC

)()( tAutx =

cbss

csH

++=

)()(

)(lim tyt →

PROPERTIES

201

202

OUTLINE

• Introduction





203

INVERSE LAPLACE TRANSFORM

• Inverse Laplace transform

01

1

1

01

1

1)(asasasa

bsbsbsbsX

n

n

n

n

m

mmm

++++

++++=

−

−

−

−

– Evaluation of the above integral requires the use of contour

integration in the complex plan ➔ difficult.

• Inverse Laplace transform: special case

– In many cases, the Laplace transform can be expressed as a

rational function of s

– Procedure of Inverse Laplace Transform

• 1. Partial fraction expansion of X(s)

• 2. Find the inverse Laplace transform through Laplace

transform table.

+

−=

j

jdsstsX

jtx

)exp()(

2

1)(

204


• Review: Partial Fraction Expansion with non-repeated linear

factors

321

)(as

C

as

B

as

AsX

−+

−+

−=

1

)()( 1 assXasA

=−=

2

)()( 2 assXasB

=−=

3

)()( 3 assXasC

=−=

• Example

– Find the inverse Laplace transform of sss

ssX

43

12)(

23 −+

+=

205


• Example

– Find the Inverse Laplace transform of

23

2)(

2

2

++=

ss

ssX

• If the numerator polynomial has order higher than or equal to the order

of denominator polynomial, we need to rearrange it such that the

denominator polynomial has a higher order.

206


• Partial Fraction Expansion with repeated linear factors

( ) bs

B

as

A

as

A

bsassX

−+

−+

−=

−−= 1

2

2

2 )()(

1)(

( )as

sXasA=

−= )(2

2 ( ) as

sXasds

dA

=

−= )(2

1( )

bssXbsB

=−= )(

207


• High-order repeated linear factors

bs

B

as

A

as

A

as

A

bsassX

N

N

N −+

−++

−+

−=

−−=

)()()()(

1)(

2

21

( )bs

sXbsB=

−= )(

( ) as

N

kN

kN

k sXasds

d

kNA

=

−

−

−−

= )()!(

1Nk ,,1=

208

OUTLINE

• Introduction




• Applications of Laplace Transform

209

APPLICATION: LTI SYSTEM REPRESENTATION

• LTI system

– System equation: a differential equation describes the input output

relationship of the system.

)()()()()()()( 0

)1(

1

)(

0

)1(

1

)1(

1

)( txbtxbtxbtyatyatyaty M

M

N

N

N +++=++++ −

−

=

−

=

=+M

m

m

m

N

n

n

n

N txbtyaty0

)(1

0

)()( )()()(

– S-domain representation

)()(0

1

0

sXsbsYsasM

m

m

m

N

n

n

n

N

=

+

=

−

=

– Transfer function

−

=

=

+

==1

0

0

)(

)()(

N

n

n

n

N

M

m

m

m

sas

sb

sX

sYsH

210


• Simulation diagram (first canonical form)

Simulation diagram

211


• Example

– Show the first canonical realization of the system with transfer function

6116

23)(

23

2

+++

+−=

sss

ssSH

212

APPLICATION: COMBINATIONS OF SYSTEMS

• Combination of systems

– Cascade of systems

– Parallel systems

)()()( 21 sHsHSH =

)()()( 21 sHsHSH +=

213


• Example

– Represent the system to the cascade of subsystems.

6116

23)(

23

2

+++

+−=

sss

ssSH

214


• Example:

– Find the transfer function of the system

LTI system

215


• Poles and zeros

)())((

)())(()(

11

11

pspsps

zszszssH

NN

MM

−−−

−−−=

−

−

– Zeros:

– Poles:

Mzzz ,,, 21

Nppp ,,, 21

216

APPLICATION: STABILITY

• Review: BIBO Stable

– Bounded input always leads to bounded output

+

−dtth |)(|

• The positions of poles of H(s) in the s-domain

determine if a system is BIBO stable.

N

N

m ss

A

ss

A

ss

AsH

−++

−+

−=

)()(

2

2

1

1

– Simple poles: the order of the pole is 1, e.g.

– Multiple-order poles: the poles with higher order. E.g.

1s Ns

2s

217


• Case 1: simple poles in the left half plane

kk jp −=1

0k( ) 22

1

kks +−

)()sin()exp(1

)( tuttth kk

k

k

=

))((

1

kkkk jsjs −−+−=

kk jp +=2

=+

−dtthk )(

• If all the poles of the system are on the left half plane,

then the system is stable.

Impulse response

218


• Case 2: Simple poles on the right half plane

kk jp +=1

0k( ) 22

1

kks +− ))((

1


kk jp −=2

)()sin()exp(1

)( tuttth kk

k

k

=

• If at least one pole of the system is on the right half

plane, then the system is unstable.

Impulse response

219


• Case 3: Simple poles on the imaginary axis

)()sin(1

)( tutth k

k

k

=

0=k( ) 22

1

kks +− ))((

1


• If the pole of the system is on the imaginary axis, it’s

unstable.

220


• Case 4: multiple-order poles in the left half plane

)()sin()exp(1

)( tutttth kk

m

k

k

= 0k stable

• Case 5: multiple-order poles in the right half plane

)()sin()exp(1

)( tutttth kk

m

k

k

= 0k

0k 0k

unstable

• Case 6: multiple-order poles on the imaginary axis

)()sin(1

)( tuttth k

m

k

k

= unstable

221


• Example:

– Check the stability of the following system.

136

23)(

2 ++

+=

ss

ssH


ELEG 3124 Signals & Systems

Ch. 6 Discrete-Time System

Dr. Jingxian Wu

[email protected]

223

OUTLINE

• Discrete-time signals

• Discrete-time systems

• Z-transform

224

SIGNAL

• Discrete-time signal

– The time takes discrete values

=

4cos)(

nnx

=

4exp

2

1)(

nnx

225

SIGNAL: CLASSIFICATION

• Energy signal v.s. Power signal

– Energy:

−=

→=

N

NnN

nxE2

)(lim

– Power:

−=

→ +=

N

NnN

nxN

P2

)(12

1lim

– Energy signal: E

– Power signal: P

226

SIGNAL: CLASSIFICATION

• Periodic signal v.s. aperiodic signal

– Periodic signal

• The smallest value of N that satisfies this relation is the fundamental

periods.

– Is periodic?

)()( Nnxnx +=

– Example: )3cos( n

)cos( n

)cos( n

)4

3cos( n

)cos( n is periodic if is integer for integer k.

k2

227

SIGNAL: ELEMENTARY SIGNAL

• Unit impulse function

==

.0

,0

,0

,1)(

n

nn

• Unit step function

=

.0

,0

,1

,0)(

n

nnu

• Relation between unit impulse function and unit step function

)1()()( −−= nunun

−=

=n

k

knu )()(

228

SIGNAL: ELEMENTARY SIGNAL

• Exponential function

)exp()( nnx =

• Complex exponential function

)sin()cos()exp()( 000 njnnjnx +==

229

OUTLINE



• Z-transform

230

SYSTEM: IMPULSE RESPONSE

• Impulse response of LTI system

– The response of the system when the input is )(n

)()( nnx =System

)()( nhny =

• System response for arbitrary input

– Any signal can be decomposed as the sum of time-shifted impulses

)()()( knkxnxk

−= +

−=

)( kn−System

)( knh −– Time invariant

– Linear

)()( knkxk

−+

−=

System

)()( knhkxk

−+

−=

LTI system

LTI system

LTI system

231

SYSTEM: CONVOLUTION SUM

• Convolution sum

– The convolution sum of two signals and is )(nx )(nh

)()()()( knhkxnhnxk

−= +

−=

• Response of LTI system

– The output of a LTI system is the convolution sum of the input and

the impulse response of the system.

)(nx)(nh

)()( nhnx

LTI system

232


• Example

– 1. )()( mnnx −

– 2. ),()( nunx n= )()( nunh n=

= )()( nhnx

233


• Example:

– Let be two

sequences, find

],1,1,0,2,1[)( −=nh]2,1,3,1[)( −−=nx

)()( nhnx

234

STSTEM: COMBINATION OF SYSTEMS

• Combination of systems

➔

+ ➔

Two systems in series

Two systems in parallel

235

SYSTEM: DIFFERENCE EQUATION REPRESENTATION

• Difference equation representation of system

==

−=−M

k

k

N

k

k knxbknya00

)()(

236

OUTLINE



• Z-transform

237

Z-TRANSFORM

• Bilateral Z-transform

n

n

znxzX −+

−=

= )()(

• Unilateral Z-transform

n

n

znxzX −+

=

=0

)()(

• Z-transform:

– Ease of analysis

– Doesn’t have any physical meaning (the frequency domain

representation of discrete-time signal can be obtained through

discrete-time Fourier transform)

– Counterpart for continuous-time systems: Laplace transform.

238

Z-TRANSFORM

• Example: find Z-transforms

– 1. )()( nnx =

– 2. )(2

1)( nunx

n

=

239

Z-TRANSFORM

• Example

– 3. )1(

2

1)( −−

−= nunx

n

• Region of convergence (ROC)

Region of convergence

240

Z-TRANSFORM: CONVERGENCE

• Convergence of causal signal

)()( nunx n=

• Convergence of anti-causal signal

)1()( −−= nunx n

Z-TRANSFORM: TIME SHIFTING PROPERTY

• Time Shifting

– Let be a causal sequence with the Z-transform

– Then

241

)(nx )(zX

−

=

−−=+1

0

0

0

00 )()()(n

m

mnnzmxzzXznnxZ

−

−=

−−−+=−

1

0

0

00 )()()(nm

mnnzmxzzXznnxZ

242

Z-TRANSFORM: LTI SYSTEM

• LTI System

– Difference equation representation

= =

−=−N

k

M

k

kk knxbknya0 0

)()(

– Z-domain representation

)()(00

zXzbzYzaM

k

k

k

N

k

k

k

=

=

−

=

−

– Transfer function

==

=

−

=

−

N

k

k

k

M

k

k

k

za

zb

zX

zYzH

0

0

)(

)()(

243

Z-TRANSFORM: LTI SYSTEM

• Example

– Find the transfer function of the system described by the following

difference equation

)1(2

1)()2(2)1(2)( −+=−+−− nxnxnynyny

244

Z-TRANSFORM: STABILITY

• Stability

az

zzH

−=)( )()( nuanh n=

– A LTI system is BIBO stable is all the poles are within the unit

circle (|a| < 1)

– A LTI system is unstable is at least one pole is on or outside of the

unit circult ( )1|| a

ELEG 3124 SYSTEMS AND SIGNALS Lecture Notes

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